For many (most) solids dissolving in water, the solubility (maximum concentration available in equilibrium) increases with temperature. But things are just the opposite with gases dissolving in water (based solutions) - the solubility decreases with increasing temperature. This is why you need higher pressures to achieve a specific target carbonation level at higher temps vs. at lower temps. For example to obtain 2.5 volumes at 70°F requires a CO2 gauge pressure of 30.4 psi (45.1 psi CO2 partial pressure), but at 35°F you only need 9.8 psi gauge (24.5 psi partial pressure) to achieve the same 2.5 volumes.
Volumes is an odd ball unit of measure, only used in the context of carbonated beverages (AFAIK), but it is simply just another scale for dissolved CO2 concentration. 1 volume of carbonation is equal to 1.977 g/L of dissolved CO2, so 2.5 volumes is equal to 1.977 g/L * 2.5 = 4.94 g/L of dissolved CO2. So, beer at 35°F carbed to 2.5 volumes contains 4.94 g/L of CO2, and beer at 70°F carbed to 2.5 volumes also contains 4.94 g/L of dissolved CO2.
Any differential of total CO2 required to carbonate a given volume of beer to a specific carbonation level, is due only to the difference in the amount of CO2 in the headspace above the beer, which varies depending on the temperature at which carbonation occurs. Since higher pressures are required at higher temperatures, there will be more CO2 in the headspace at higher temperatures. This is the calculation I did in a post further up this thread.
Brew on